Abstract
We study the four-dimensional mathcal{N} = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge mathcal{Q} , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding mathcal{N} = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.
Highlights
The coefficients hO furnish the basic structure constants for the CFT on RPd
We find that for a large class of local and extended SYM observables preserving a common supercharge Q, their expectation values are captured by an effective two-dimensional bosonic YangMills (YM) theory on an RP2 submanifold
A large class of CFTs are produced by renormalization group (RG) flows from four-dimensional Yang-Mills theories coupled to matter
Summary
On flat space R4 with coordinates xμ, the real projective space RP4 is defined by identifying points related by a fixed-point-free involution ι. To define the N = 4 SYM on RP4 preserving the half-BPS supersymmetry, we require the following supersymmetric identification due to ιSYM. The residual (superconformal) symmetry on RP4 with round metric (2.17) is given by the Weyl transformation of the flat space counterparts. Which follows from the identification (2.15) on R4 after taking into account the Weyl factors in (2.19) We emphasize that this identification is a symmetry of the action (2.26), and respects the off-shell SUSY transformation laws (2.27), provided that we implement the same identification for the auxiliary pure spinors as in (2.9) (with the Weyl factor in (2.17)
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